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Hello Mathoverflow Community,

I would really appreciate some advice on this:

All I know is basic calculus and basic linear algebra, I want to start learning more advanced material on my own while taking more advanced calculus/ linear algebra courses.

Is there any area of mathematics which I can delve into with only this much knowledge? (ex: topology, number theory, etc.) or should I instead fully focus on my courses for now?

Thank you very much,


Thank you so much for all of your comments

Yes I am a freshman in university, and by basic I meant Calculus I, II, and (now) III, and I'm in a linear algebra I course. I find myself really good at calculus, I pick up new topics really fast. However, I'm still improving in linear algebra.

I have picked up a couple of books on proofs, I seem to be doing well with it, However I exposed myself to a "Elementary number theory" book and I felt like a bit of background is missing (especially in understanding advanced proofs).

Thank you once again for your amazing advice and comments, it really means a lot to me to get such advice at this stage.

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    $\begingroup$ You should fully focus on your courses right now. If you find that you have spare time that you want to spend doing mathematics, you can pick up books and learn subjects in the order in which you would learn them in your course. Basically, the reason things are taught in the order they are taught in is because many very clever and knowledgeable people thought about the best order to teach things in and the result is the modern university curriculum. You will not do yourself a favour by skipping, say, introductory abstract algebra and trying to learn class field theory instead. $\endgroup$
    – Alex B.
    Dec 16 '10 at 5:03
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    $\begingroup$ One more thing: you shouldn't stress too much about getting to the forefront of modern research as quickly as possible. That would only slow you down in the long run. Just learn at a pace at which you enjoy the maths. Not only will that make it easier for you to make an informed decision about what you actually want to do, you will also get the frontiers much more quickly than might seem to you at the moment. In particular, if you do decide to read Artin's Algebra, don't try to swallow it. Do all the exercises and make sure you feel comfortable with the concepts. $\endgroup$
    – Alex B.
    Dec 16 '10 at 5:10
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    $\begingroup$ You can start working on open problems in, say, combinatorial probability right away (Take a graph on $n$ vertices. Do $100n$ steps randomly. Show that the probability to visit all vertices is exponentially small in $n$ for large $n$. Nobody knows how to do it, and you don't need any advanced knowledge to start tinkering with it). Third, make sure that, while you are getting your education, you do research in addition to regular studies, not instead of them. Fourth, talk to your professors about research options (works only if you pass all exams with A's) $\endgroup$
    – fedja
    Dec 16 '10 at 5:37
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    $\begingroup$ @Alex: I'm very uncomfortable with that appeal to authority. Just because the OP's university does things in some particular way doesn't mean that there are also very smart people who do things another way. For instance, if the OP is at a school where no sort of honors math/math for math majors specific course, then your advice would probably be detrimental to the OP's mathematical development. $\endgroup$ Dec 16 '10 at 6:56
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    $\begingroup$ I also disagree with Alex. Many people, including myself, learn better by pushing ahead to see what lies ahead of us, and then going back to fill in the earlier material when we understand why it matters. You shouldn't push so far ahead that you lose track of your current courses but, if you are solving almost all the problems on your problem sets, and you are excited about getting a glimpse of higher math, go for it! $\endgroup$ Dec 16 '10 at 15:33

12 Answers 12

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Stillwell's Naive Lie theory was essentially written as an answer to this question. I quote from the introduction:

It seems to have been decided that undergraduate mathematics today rests on two foundations: calculus and linear algebra. These may not be the best foundations for, say, number theory or combinatorics, but they serve quite well for undergraduate analysis and several varieties of undergraduate algebra and geometry. The really perfect sequel to calculus and linear algebra, however, would be a blend of the two —a subject in which calculus throws light on linear algebra and vice versa. Look no further! This perfect blend of calculus and linear algebra is Lie theory (named to honor the Norwegian mathematician Sophus Lie—pronounced “Lee ”).

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    $\begingroup$ I'm no expert here, but it would seem to me that calculus and linear algebra are an excellent foundation for combinatorics, especially at the advanced undergraduate level. I took exactly one course on combinatorics, from the great Laszlo Babai, and the tools we used were indeed calculus (e.g. knowledge of asymptotics of functions and the ability to optimize certain constructions) and linear algebra (in very clever ways as in the book on the subject by Babai and Frankl). And I don't view the fact that you could probably get away with even less than this as detracting from my assertion. $\endgroup$ Dec 16 '10 at 8:29
  • $\begingroup$ To put the assertion of my previous comment more positively: it seems to me that the OP could now study combinatorics, if he so chose. $\endgroup$ Dec 16 '10 at 8:30
  • $\begingroup$ This is indeed an excellent suggestion, Qiaochu. $\endgroup$ Dec 16 '10 at 8:32
  • $\begingroup$ @Pete: I think the operative word in that sentence is best. I certainly agree that calculus and linear algebra are quite useful in combinatorics and that the OP should feel free to study combinatorics. $\endgroup$ Dec 16 '10 at 8:44
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    $\begingroup$ Great recommendation. I never knew about this book, and it appears to be exactly the book I wish I had when I was a graduate student. $\endgroup$
    – Deane Yang
    Dec 16 '10 at 16:34
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I think if you like Taylor series, then Herb Wilf's generatingfunctionology would be a good choice. It shows how you can use Taylor series to solve counting problems in combinatorics. You can download the second edition from Wilf's homepage.

You could also just try looking at textbooks for the undergraduate math major courses, such as abstract algebra, and real analysis. Michael Artin's Algebra gives a fairly broad introduction to the subject of abstract algebra. For real analysis, many people swear by Rudin's Principles of Mathematical Analysis. There are many other texts that cover the same material.

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Dear James,

To a large extent, the answer to this question will depend on how successful you are at working on your own without the infrastructure of a course/lecturer/problem sets/etc. to guide you. Given this, if you don't know yet whether you work well by yourself, there's only one way to find out: try it! You may find that you are good at working by yourself, and, if so, it doesn't really matter what your background is: you can fill it in by reading more books. On the other hand, you may find that it's hard to make progress without the usual structures that a course provides, and that's fine; many successful mathematicians were not all that independent when they were undergraduates.

One book that you can read which doesn't require much background at all is Hardy and Wright's classic text on number theory. It does not suit everyone's taste, but if you are not yet sure where your taste lies, you can take a look and see if you like it.

One thing that you didn't address in your post is the question of how comfortable you are with reading and writing proofs. If you are not comfortable with this aspect of mathematics, then my suggestion of Hardy and Wright won't be terribly appropriate, and neither will many of the others. If you are comfortable with proofs, then in some sense there is no limit on what you can do by yourself, since (at least in principle) you can pick up any textbook and try to learn what is in it. On the other hand, if you find that you aren't (yet) comfortable with reading and understanding formal proofs by yourself, then it will be harder to go very far by yourself, and it might be better to focus on your formal course work for now. (And, if your ambition is to pursue pure mathematical research, you should try to take courses that introduce you to reading and writing proofs as soon as you can.)

Whatever your situation is, you should always be sure not to neglect your formal coursework (even if the work you are doing on your own turns out to be more exciting). Excellence in formal coursework is more or less a requirement for going on in graduate school, which is in turn a requirement for becoming a research mathematician.

Regards,

Matthew

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I will say: many areas and not a lot. Ha!

Let me explain: one the one hand, linear algebra and calculus are enough to consider a lot of non-trivial problems and describe basic issues in many areas. On the other hand, the various areas of mathematics tend to interact intensely with each other, which is what makes math so cool. So it's going to be difficult to direct you to a specific area, since chances are that a reference that is advanced enough will not be shy about using much more advanced notions (check out the math articles on wikipedia to get an idea of what I mean; even innocuous sounding ones can get pretty intense).

I do want to encourage you to give in to your curiosity: but instead of picking a specific subject, you would be much better off picking up specific references that are written more specifically for your level. There are many of those, look for general math books, e.g. from the AMS and MAA. "Proofs from THE BOOK" might be a bit intense, but roughly at the right level.

Since the various areas of math tend to riff off each other as I mentioned, the last thing you want to do is get specialized too early anyway, so generalist books are better for you now.

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    $\begingroup$ Since you mention books from the AMS and MAA, I'll point out that the Student Mathematical Library series (STML) from the AMS has many books targeted at the level of undergraduates who are ready to move beyond calculus and linear algebra and are looking for something suitable for self-study. There are other book series as well, some of which are similarly targeted, so the OP may find something that tickles his fancy browsing somewhere in there. $\endgroup$ Dec 16 '10 at 18:01
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I think a nice and interesting topic that seems doable with basic calculus and linear algebra would be some kind of introduction in the theory of knots and surfaces. In particular, I have in mind the book "Knots and Surfaces. A guide to discovering mathematics" by Gilbert and Porter, Oxford Univ. Press, 1995. I think it would not be too sophisticated for you, it will introduce you to and get you thinking about various important objects in mathematics, and it may inspire you for your later studies. Have a look at it. If it turns out to be not that well doable for you, you could always take a second look at it in a year or so.

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    $\begingroup$ Other books along these lines: Stillwell's Geometry of Surfaces, Adams' The Knot Book, Weeks' The Shape of Space (no rigorous definitions, but lots of interesting pictures and exercises). $\endgroup$ Dec 16 '10 at 21:27
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@James, OP of this fine question:

I've edited this answer in light of your response. Thanks for getting back to us with the details of your mathematical education to this point. As you can see from one of my comments, I was a little concerned that you might have forgotten us! In any event, my follow up is presented in the paragraph after this next one, which I'm leaving in as part of my original answer to this question.

My original response was:

I think, if you want to get "better"answers--by which I mean answers more precisely tailored to your individual level of mathematical development, I think it would help if you edited your question (since you can't make comments until you have 50 reputation points) so as to specify exactly what you mean by "basic". It sounds to me like you have been exposed to single-variable calculus and linear algebra through maybe determinants. To offer a few hints as to what I'm fishing for here, perhaps you could tell us if you have studied: a.) infinite series; b.) partial derivatives and multiple integrals; c.) eigevalues and eigenvectors; d.) characteristic polynomials of matrices; e.) the Hamilton-Cayley theorem; f.) vector calculus--gradient, divergence and curl; g.)linear ordinary differential equations. If you do that, I'll try to answer your question. (You can find my email address on my user profile in case I forget to check back.) Meanwhile, Qiaochu Yuan's answer looks fascinating to me, as does the problem fedja pitched.

And my addenda are:

First of all, it sounds to me like you have encountered, or are about to encounter, almost everything I mentioned in your course work. Let's see, you've had a full year of calculus, if I understand you, and you are in the first half of your second year. So if your courses are anything like mine were, you have probably seen items (a.) and (b.) on my list--you are probably just getting into partial derivatives etc. right about now. I would guess you've scratched the surface of item (f.), and probably have been exposed to eigenvalues and eigenvectors (item (c.)), and perhaps the characteristic polynomial (item (d.)). I'd bet that items (e.) and (g.) are just up the road in your course work. That being said, I think there are a few really good books you could probably tackle without too much difficulty. First of all, you might check out the book Differetial Equations, Dynamical Systems and and Introduction to Chaos by Hirsh, Smale and Devaney. This is an introductory text on differential equations which includes some very nice explanations of some fairly advanced topics; it should be pretty accessible to a person with your background. If you are interested in abstract algebra, you might have a look at Emil Artin's little book called Galois Theory; it covers some central material on groups and fields, right from the ground up. Incidentally, Smale, Hirsh and Devaney explains most of the linear algebra needed as you go along, so anything you haven't seen will be covered. If you like topology, and are ready for a challenge, you might look into John Milnor's Topology from the Differentiale Viewpoint. Finally, Barrett O'Neill's Elementary Differential Geometry covers the basics of this field, and as I recall only requires knowledge of calculus at your level, plus some linear algebra. All these books are good introductions to topics of great interest to many mathematicians at the present time.

Don't forget to try the problems--math is like music; you've got to practice.

Good luck with it! Let us know how it goes!

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Combinatorics! Some deep results can sometimes be proved or simplified by finding the right objects to count. For example, the RSK algorithm is used to prove several non-trivial identities, but RSK is not really that hard to understand.

Also, if you are a decent programmer, you can probably attack several unsolved problems right now, by looking at some special cases and guessing patterns.

Some open questions in combinatorics right now are really about guessing the right object/formula, proving the formula is usually (hopefully) easier than finding it.

Proof by induction, inclusion-exclusion, sign-reversing involutions and generating functions and good intuition are the main tools, rather than lots of deep theorems.

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Pressley's Elementary Differential Geometry requires little more than some multivariable calculus and linear algebra. It treats curves and surfaces in $\mathbb{R}^{3}$. However, the author is careful to point out that while many of the results generalise to higher dimensions, the methods used in the book do not always do so. This is done in part to make the subject accessible. It might be worth a look to get a taste of differential geometry without the machinery developed in more advanced courses on topology, smooth manifolds and the like.

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    $\begingroup$ Petrunin's book on github (anton-petrunin.github.io/papers) is cheaper (free) and (in my opinion) more beautiful and more beneficial for the student. $\endgroup$
    – Ben McKay
    Aug 1 at 17:06
  • $\begingroup$ @BenMcKay, thank you very much for pointing out Petrunin's book. $\endgroup$
    – J W
    Aug 1 at 17:17
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A linear algebra point of view can be useful for some topics normally addressed in a second-year calculus course. Understanding jacobians and doing some things with systems of differential equations that require eigenvectors, etc.

Then in statistics, suppose you want to understand why the sum of squares of residuals in a simple linear regression problem has a scalar multiple of a chi-square distribution with $n-2$ degrees of freedom, where $n$ is the number of data points, and why it's independent of the estimate of the slope. That all becomes clear if you know how a real symmetric matrix can be diagonalized by an orthogonal matrix. Or suppose you want to understand why every non-negative definite symmetric real matrix can be realized as the variance of some random vector. Same thing.

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    $\begingroup$ Another small comment on statistics. Anyone with a pseudo-random number generator and some bits of commonplace software can draw a (pseudo-)random sample of 200 observations from a bivariate normal distribution with correlation (for example) 0.8. The correlation in the sample is on average about 0.8, but from one sample to the next varies with a standard deviation of about 0.025. So if you draw such a sample and get 0.825 or 0.775 as the correlation, that's unsurprising. What if you want to sample from the conditional distribution given the correlation, so that..... $\endgroup$ Dec 16 '10 at 18:16
  • $\begingroup$ .....you always get exactly 0.8, while the 200 points continue to vary randomly from one sample of 200 to the next? I wrote a little program that does that. I did it by knowing about spectral decompositions of 2-by-2 matrices. The theory of the multivariate normal distribution, the F-distribution, Student's t-distribution, the chi-square distribution, the Wishart distribution, etc., requires linear algebra to be understood. If you've got linear algebra and first-year calculus, you can do quite a lot of that sort of thing. $\endgroup$ Dec 16 '10 at 18:20
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I would highly recommend trying out mathematical logic and maybe also some introductory set theory. Logic is more or less self-contained and learning how to write up formal proofs is essential in any higher level of mathematics that you encounter. The nice thing about working in logic is that it trains you to formally prove that which is often intuitively clear. The same goes for proofs in finite set theory. But with set theory, you can also quickly work up to some results that are often initially counterintuitive involving the infinite.

Perhaps someone else can recommend some references here since my pre-college/undergrad knowledge in these areas came from a variety of sources including oral presentations and course packets.

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  • $\begingroup$ I quite liked Thomas Forster's Logic, induction and sets — it has the right mix of philosophy and mathematics. It's also somewhat non-standard since he emphasises the notion of recursive types, which gives it a bit of a computer-science sort of flavour. I also advise some caution — a formal proof in the sense of logic, is quite different to a formal proof in the sense of mathematics! (Indeed, it's more akin to the so-called two-column proofs from high school, if I understand the descriptions correctly.) $\endgroup$
    – Zhen Lin
    Dec 20 '10 at 14:07
  • $\begingroup$ @Zhen, there are definitely two-column proofs in logic, but there is so much more. To name a few examples, you have the Compactness Theorem, Gödel's Completeness theorem, and Gödel's Incompleteness theorems. You also have all of the machinery involved in the proofs of the latter two results. Also, all of our proofs are theoretically supposed to be able to be transformed into a two-column proof to verify correctness, but I don't see that happening in practice anytime soon. $\endgroup$
    – Jason
    Dec 20 '10 at 19:16
  • $\begingroup$ @Jason I'm quite aware of that — I was simply making the amusing observation that the object "formal proof" studied in mathematical logic is, oxymoronically, precisely not the kind of "formal proof" we mean in the rest of mathematics (including in logic itself)! $\endgroup$
    – Zhen Lin
    Dec 21 '10 at 7:42
  • $\begingroup$ @Zhen: Yes, we definitely would not want to translate Wiles's proof of Fermat's Last Theorem into a "formal proof". :) Also, my comment was intended to clarify to the OP the value of learning logic. I want to apologize for the fact that my wording suggested you weren't aware of the theorems in logic not proved with the two-column proof. $\endgroup$
    – Jason
    Dec 21 '10 at 9:22
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The problem is that Calculus has little-to-no proof content (unless your course used a book like Spivak's). And depending on what kind of Linear Algebra course you took, it was either mostly computational and had little proof content, or it was used to introduce you to proofs. But even in the latter scenario, you wouldn't have enough exposure to proofs yet to easily jump into an advanced subject.

Proofs are the lingua franca of mathematics, but most students have to take multiple proof-based courses to develop even a moderate amount of comfort with mathematical rigor. And without understanding what makes a proof rigorous, you can't really do anything interesting in pure math.

The kind of math you learn before college is mostly "plug and chug". Someone gives you a recipe for how to solve a problem, and you apply that recipe. So long as you apply the recipe correctly, you get the correct result. This type of "math" continues into early college-level courses such as Calculus. In Calculus, it may not always be immediately obvious what recipe you need to apply to a given problem, but it's still mainly about combining a few recipes and some other tricks you've learned. With Linear Algebra, even if you're doing a proof-based Linear Algebra course, most of the proofs primarily revolve around algebraic manipulation techniques.

This way of looking at math -- as a tool that gives you a solution -- is useful if you want to study an applied area that uses mathematics, but not if you want to study mathematics itself. For instance, with Calculus and Linear Algebra, you know enough to dive into an area like Machine Learning or Computer Graphics, or probably several Engineering fields.

But studying mathematics means getting a deeper understanding of why the tools work, not just how to use them. It requires a very different mindset than what you have been taught so far.

The best recommendation would be to look at a book on elementary number theory, which it sounds like you already attempted to do. Number theory has a ton of problems that are easy to understand, but which don't have an easy solution. In fact, many famous problems in number theory remain unsolved. But you're not likely to be able to solve them without learning a lot more advanced mathematics.

It's not so much that understanding advanced mathematics is absolutely required to solve those problems. It's at least hypothetically possible that someone could solve these hard problems from first principles, knowing not much more than you already know. Possible but unlikely. Rome wasn't built over night, and even very brilliant people build on the foundations of those that came before them.

So the reason for studying a huge breadth of mathematics, rather than immediately jumping into some deep problem that looks interesting, is the breadth will give you more perspectives to look at the problem from, thus increasing the chance that you may figure out a creative solution.

However, many elementary number theory books are specifically tailored to students with little math background, and if you are struggling to understand the proofs in such a book, it means that you just don't have enough familiarity with proof, which will be an issue no matter what math subject you study. Your college has probably given some thought to developing a curriculum that will increase your familiarity with proof over time, so it's recommended you follow their curriculum.

Think of proof as a game. Right now, you probably don't even understand the rules of the game that well. Once you get to the point where you can play the game well enough that you are able to understand how expert players play it (even if you can't replicated what they do), then you can go study any advanced area of mathematics on your own. Until then, you need to focus on getting better at this game.

One more thought. You could go back and read Spivak's Calculus textbook. If you do so, you are likely to find the concepts familiar and foreign to you at the same time. Reading that book will make you realize that even in the subject of Calculus, there are so many important details that your previous instructor never taught you. Details that are not important if you will never major in math, but which are important if you want to truly understand the subject. And it's not like you will have to use Calculus much in the future. It's just one of many things you learn in math.

But what's important for you to understand is how a mathematician actually looks at Calculus, which is almost certainly very different than how you were taught. And the reason for that is because math professors believe their perspective is wasted on students who aren't going to become math majors. Ergo, you will not even get a good sense of how mathematicians look at math until you take the first course that only math majors take.

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Perhaps if you are analytically inclined then analysis on fractals is a good place to look. A good portion of the work done is with second finite difference equations leading towards a limit definition of a ``second derivative'' and uses some linear algebra such as inverting small matrices. Strichartz's Differential Equations on Fractals is a good place to start, especially the first few chapters where spectral decimation is discussed. As an aside signifigant number of papers in this area have come out of REU programs.

In general focusing on your courses is important but that should still leave you with some time to think about other topics as well. This kind of curiosity will help you see what's out there and give you a sensible way to choose a specialty when the time comes.

Best of luck.

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